Efficient Preconditioners Based on Fictitious Domains for Elliptic Fe{problems with Lagrange Multipliers Eecient Preconditioners Based on Ctitious Domains for Elliptic Fe{problems with Lagrange Multipliers

The macro{hybrid formulation based on domain decomposition is considered for elliptic boundary value problems with both symmetric positive deenite and indeenite operators. The problem is discretized by the mortar element method, which leads to a large{scale sparse linear system with a saddle{ point matrix. In the case of symmetric and positive deenite operators, a block diagonal preconditioner based on ctitious domains is proposed, which is spectrally equivalent to the original saddle{point matrix. In the case of the Helmholtz operator with the absorbing boundary conditions, a special preconditioner is introduced such that the subspace of constraints remains invariant with respect to the preconditioned GMRES method. Results of numerical experiments are presented.